Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

Abstract

For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the "Schrödinger conjecture" and the "Last's intersection spectrum conjecture" have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian and Jitomirskaya-Mar. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

References in corpus (5)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Complex one-frequency cocycles
• Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications
• Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles

Cited by in corpus (1)

• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles